Appleton-Hartree equation

The Appleton-Hartree equation, sometimes also referred to as the Appleton-Lassen equation is a mathematical expression that describes the refractive index for electromagnetic wave propagation in a cold magnetized plasma. The Appleton-Hartree equation was developed independently by several different scientists, including Edward Victor Appleton, Douglas Hartree and K. Lassen.

Equation

Full Equation

The equation is typically given as follows [ Citation | last = Helliwell | first = Robert | date = 2006 | title = Whistlers and Related Ionospheric Phenomena | edition = 2nd | publication-place = Mineola, NY | publisher = Dover | pages = 23-24 ] :

:$n^2\; =\; 1\; -\; frac\{X\}\{1\; -\; iZ\; -\; frac\{frac\{1\}\{2\}Y^2sin^2\; heta\}\{1\; -\; X\; -\; iZ\}\; pm\; frac\{1\}\{1\; -\; X\; -\; iZ\}left(frac\{1\}\{4\}Y^4sin^4\; heta\; +\; Y^2cos^2\; hetaleft(1\; -\; X\; -\; iZ\; ight)^2\; ight)^\{1/2$

Definition of Terms

$n$ = complex refractive index

$i$ = $sqrt\{-1\}$

$X\; =\; frac\{omega\_0^2\}\{omega^2\}$

$Y\; =\; frac\{omega\_H\}\{omega\}$

$Z\; =\; frac\{\; u\}\{omega\}$

$u$ = electron collision frequency

$omega\; =\; 2pi\; f$

$f$ = wave frequency

$omega\_0\; =\; 2pi\; f\_0\; =\; sqrt\{frac\{Ne^2\}\{epsilon\_0\; m$ = electron plasma frequency

$omega\_H\; =\; 2pi\; f\_H\; =\; frac\{B\_0\; |e\{m\}$ = electron gyro frequency

$epsilon\_0$ = permittivity of free space

$mu\_0$ = permeability of free space

$B\_0$ = ambient magnetic field strength

$e$ = electron charge

$m$ = electron mass

$heta$ = angle between the ambient magnetic field vector and the wave vector

Modes of Propagation

The presence of the $pm$ sign in the Appleton-Hartree equation gives two separate solutions for the refractive index [ Citation | last = Bittencourt | first = J.A. | date = 2004 | title = Fundamentals of Plasma Physics | edition = 3rd | publication-place = New York, NY | publisher = Springer-Verlag | pages = 419-429 ] . For propagation perpendicular to the magnetic field, i.e., $kperp\; B\_0$, the '+' sign represents the "ordinary mode," and the '-' sign represents the "extraordinary mode." For propagation parallel to the magnetic field, i.e., $kparallel\; B\_0$, the '+' sign represents a left-hand circularly polarized mode, and the '-' sign represents a right-hand circularly polarized mode. See the article on electromagnetic electron waves for more detail.

Reduced Forms

Propagation in a Collisionless Plasma

If the wave frequency of interest $omega$ is much smaller than the electron collision frequency $u$, the plasma can be said to be "collisionless." That is, given the condition

$u\; ll\; omega$,

we have

$Z\; =\; frac\{\; u\}\{omega\}\; ll\; 1$,

so we can neglect the $Z$ terms in the equation. The Appleton-Hartree equation for a cold, collisionless plasma is therefore,

:$n^2\; =\; 1\; -\; frac\{X\}\{1\; -\; frac\{frac\{1\}\{2\}Y^2sin^2\; heta\}\{1\; -\; X\}\; pm\; frac\{1\}\{1\; -\; X\}left(frac\{1\}\{4\}Y^4sin^4\; heta\; +\; Y^2cos^2\; hetaleft(1\; -\; X\; ight)^2\; ight)^\{1/2$

Quasi-Longitudinal Propagation in a Collisionless Plasma

If we further assume that the wave propagation is primarily in the direction of the magnetic field, i.e., $heta\; approx\; 0$, we can neglect the $Y^4sin^4\; heta$ term above. Thus, for quasi-longitudinal propagation in a cold, collisionless plasma, the Appleton-Hartree equation becomes,

:$n^2\; =\; 1\; -\; frac\{X\}\{1\; -\; frac\{frac\{1\}\{2\}Y^2sin^2\; heta\}\{1\; -\; X\}\; pm\; Ycos\; heta\}$

References

;Citations and notes

ee also

Plasma (physics)